Showing papers in "Siam Journal on Mathematical Analysis in 2008"
TL;DR: The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R.], with the basic proofs in [Proceedings of the Narvi...
Abstract: The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs in [Proceedings of the Narvi...
470 citations
TL;DR: A new entropy-like inequality introduced by Bresch and Desjardins for the shallow water system of equations gives additional regularity for the density (provided such regularity exists at initial time) and it is proved that the solution is unique in the class of weak solutions satisfying the usual entropy inequality.
Abstract: We consider Navier–Stokes equations for compressible viscous fluids in one dimension. It is a well-known fact that if the initial datum are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists locally in time. In this paper, we show that under the same hypothesis, the density remains bounded by below by a positive constant uniformly in time, and that strong solutions therefore exist globally in time. Moreover, while most existence results are obtained for positive viscosity coefficients, the present result holds even if the viscosity coefficient vanishes with the density. Finally, we prove that the solution is unique in the class of weak solutions satisfying the usual entropy inequality. The key point of the paper is a new entropy-like inequality introduced by Bresch and Desjardins for the shallow water system of equations. This inequality gives additional regularity for the density (provided such regularity exists at initial time).
185 citations
TL;DR: The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves and to the long-time dynamics of solutions of NLSs.
Abstract: Nonlinear Schrodinger equations (NLSs) with focusing power nonlinearities have solitary wave solutions The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves and to the long-time dynamics of solutions of NLSs We study these spectra in detail, both analytically and numerically
179 citations
TL;DR: The solutions of this unsteady fluid-structure interaction problem are studied, as the coefficient modeling the viscoelasticity of the plate tends to zero and the existence of at least one weak solution for the limit problem (Navier–Stokes equation coupled with a plate in flexion) is obtained.
Abstract: We consider a three-dimensional viscous incompressible fluid governed by the Navier–Stokes equations, interacting with an elastic plate located on one part of the fluid boundary. We do not neglect the deformation of the fluid domain which consequently depends on the displacement of the structure. The purpose of this work is to study the solutions of this unsteady fluid-structure interaction problem, as the coefficient modeling the viscoelasticity (resp., the rotatory inertia) of the plate tends to zero. As a consequence, we obtain the existence of at least one weak solution for the limit problem (Navier–Stokes equation coupled with a plate in flexion) as long as the structure does not touch the bottom of the fluid cavity.
169 citations
TL;DR: It is shown that the perturbation theory for dual semigroups (sun-star-calculus) is equally efficient for dealing with Volterra functional equations and both the stability and instability parts of the principle of linearized stability and the Hopf bifurcation theorem are obtained.
Abstract: We show that the perturbation theory for dual semigroups (sun-star-calculus) that has proved useful for analyzing delay-differential equations is equally efficient for dealing with Volterra functional equations. In particular, we obtain both the stability and instability parts of the principle of linearized stability and the Hopf bifurcation theorem. Our results apply to situations in which the instability part has not been proved before. In applications to general physiologically structured populations even the stability part is new.
150 citations
TL;DR: It is shown that the spreading speed is linearly determinate and coincides with the minimal wave speed of traveling waves.
Abstract: The spreading speeds and traveling waves are established for a class of nonmonotone discrete-time integrodifference equation models. It is shown that the spreading speed is linearly determinate and coincides with the minimal wave speed of traveling waves.
144 citations
TL;DR: In this article, a new proof of strong displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound is given.
Abstract: In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto and Westdickenberg [SIAM J. Math. Anal., 37 (2005), pp. 1227–1255] and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.
133 citations
TL;DR: It is proved the existence of global weak solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients when the initial data are large and spherically symmetric by constructing suitable aproximate solutions.
Abstract: We prove the existence of global weak solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients when the initial data are large and spherically symmetric by constructing suitable aproximate solutions. We focus on the case where those coefficients vanish on vacuum. The solutions are obtained as limits of solutions in annular regions between two balls, and the equations hold in the sense of distribution in the entire space-time domain. In particular, we prove the existence of spherically symmetric solutions to the Saint–Venant model for shallow water.
123 citations
TL;DR: This paper considers the problem of scattering of time-harmonic acoustic waves by a bounded, sound soft obstacle in two and three dimensions, studying dependence on the wave number in two clas...
Abstract: In this paper we consider the problem of scattering of time-harmonic acoustic waves by a bounded, sound soft obstacle in two and three dimensions, studying dependence on the wave number in two clas...
123 citations
TL;DR: This work establishes the existence of “local” solutions and characterize this class with the aid of an adequate version of cyclical monotonicity and shows that local solutions are induced by transport maps.
Abstract: We consider the non-nonlinear optimal transportation problem of minimizing the cost functional $\mathcal{C}_\infty(\lambda) = \operatornamewithlimits{\lambda-ess\,sup}_{(x,y) \in \Omega^2} |y-x|$ in the set of probability measures on $\Omega^2$ having prescribed marginals. This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. We establish the existence of “local” solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.
117 citations
TL;DR: The Cauchy problem for the derivative nonlinear Schrodinger equation with periodic boundary condition is considered and an adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method is considered.
Abstract: The Cauchy problem for the derivative nonlinear Schrodinger equation with periodic boundary condition is considered. Local well-posedness for data $u_0$ in the space $\widehat{H}^{s}_{r}(\mathbb{T})$, defined by the norms $\|u_0\|_{\widehat{H}^{s}_{r}(\mathbb{T})} = \|\langle\xi\rangle^s\widehat{u}_0\|_{\ell^{r'}_{\xi}}$, is shown in the parameter range $s \ge \frac{1}{2}$, $2>r>\frac{4}{3}$. The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method.
TL;DR: A family of entropy-entropy dissipation inequalities is derived in arbitrary space dimensions, and rates of the exponential decay of the weak solutions to the homogeneous steady state are estimated.
Abstract: The logarithmic fourth-order equation $\partial_t u + \frac12\sum_{i,j=1}^d\partial_{ij}^2(u\partial_{ij}^2\log u) = 0,$ called the Derrida–Lebowitz–Speer–Spohn equation, with periodic boundary conditions is analyzed. The global-in-time existence of weak nonnegative solutions in space dimensions $d\leq 3$ is shown. Furthermore, a family of entropy-entropy dissipation inequalities is derived in arbitrary space dimensions, and rates of the exponential decay of the weak solutions to the homogeneous steady state are estimated. The proofs are based on the algorithmic entropy construction method developed by the authors and on an exponential variable transformation. Finally, an example for nonuniqueness of the solution is provided.
TL;DR: In this article, it was shown that the principal eigenvalue of the linearization around zero is well defined and that a nontrivial solution of the nonlinear problem exists if and only if this eigen value is negative.
Abstract: Let $J \in C(\mathbb{R})$, $J\ge 0$, $\int_{\mbox{\tinyR}} J = 1$ and consider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. We study the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$, where f is a KPP-type nonlinearity, periodic in x. We show that the principal eigenvalue of the linearization around zero is well defined and that a nontrivial solution of the nonlinear problem exists if and only if this eigenvalue is negative. We prove that if, additionally, J is symmetric, then the nontrivial solution is unique.
TL;DR: The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg–de Vries and modified Kortweg– de Vries equations, respectively.
Abstract: In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg–de Vries-type $u_t+u^pu_x-Mu_x=0$, with M being a general pseudodifferential operator and where $p\geq1$ is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin–Ono equation. The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg–de Vries and modified Korteweg–de Vries equations, respectively. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.
TL;DR: The sequence of solutions of the original problem converges to the solution of the so-called macroscopic problem using the homogenization technique of two-scale convergence and the convergence of the nonlinear terms on the surfaces is shown.
Abstract: We study the problem of diffusive transport of biomolecules in the intercellular space, modeled as porous medium, and of their binding to the receptors located on the surface membranes of the cells. Cells are distributed periodically in a bounded domain. To describe this process we introduce a reaction-diffusion equation coupled with nonlinear ordinary differential equations on the boundary. We prove existence and uniqueness of the solution of this problem. We consider the limit, when the number of cells tends to infinity and at the same time their size tends to zero, while the volume fraction of the cells remains fixed. Using the homogenization technique of two-scale convergence, we show that the sequence of solutions of the original problem converges to the solution of the so-called macroscopic problem. To show the convergence of the nonlinear terms on the surfaces we use the unfolding method (periodic modulation). We discuss applicability of the result to mathematical description of membrane receptors ...
TL;DR: The radial symmetry of the positive solutions to the elliptic system above with critical exponents is proved and it is proved that u=v, which is a key point for the uniqueness result.
Abstract: We prove the uniqueness of the positive solutions of the following elliptic system: (1) $-\Delta(u(x))=u(x)^{\alpha}v(x)^{\beta}$, (2) $-\Delta(v(x))=u(x)^{\beta}v(x)^{\alpha}$ Here $x\in R^n$, $n\geq3$, and $1\leq\alpha<\beta\leq\frac{n+2}{n-2}$ with $\alpha+\beta=\frac{n+2}{n-2}$ In the special case when $n=3$ and $\alpha =2$, $\beta=3$, the system is closely related to the ones from the stationary Schrodinger system with critical exponents for the Bose–Einstein condensate As the first step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents We then prove that $u=v$, which is a key point for our uniqueness result
TL;DR: In this paper, the authors study the local complete synchronization of discrete-time dynamical networks with time-varying couplings and show that the Hajnal diameter of the infinite Jacobian matrices can be used to verify synchronization.
Abstract: We study the local complete synchronization of discrete-time dynamical networks with time-varying couplings. Our conditions for the temporal variation of the couplings are rather general and include variations in both the network structure and the reaction dynamics; the reactions could, for example, be driven by a random dynamical system. A basic tool is the concept of the Hajnal diameter, which we extend to infinite Jacobian matrix sequences. The Hajnal diameter can be used to verify synchronization, and we show that it is equivalent to other quantities which have been extended to time-varying cases, such as the projection radius, projection Lyapunov exponents, and transverse Lyapunov exponents. Furthermore, these results are used to investigate the synchronization problem in coupled map networks with time-varying topologies and possibly directed and weighted edges. In this case, the Hajnal diameter of the infinite coupling matrices can be used to measure the synchronizability of the network process. As ...
TL;DR: The well posedness of the Cauchy problem is proved, and some qualitative properties of the model are described.
Abstract: We present a model for the description of a nonviscous isentropic or isothermal fluid crossing a junction. Aiming at an extension of the usual Euler equations, we neglect the effects of friction against the walls of the pipes, but the reaction constraints at the junction are considered. The well posedness of the Cauchy problem is proved, and some qualitative properties of the model are described.
TL;DR: The global well-posedness of the critical dissipative quasi-geostrophic equation for large initial data belonging to the critical Besov space is proved.
Abstract: We prove the global well-posedness of the critical dissipative quasi-geostrophic equation for large initial data belonging to the critical Besov space $\dot B^0_{\infty,1}(\RR^2).$
TL;DR: For the Cauchy problem, existence, uniqueness, and Lipschitz continuous dependence of the solution from the initial data as well as from the conditions at the junction are proved.
Abstract: This paper deals with $2\times 2$ conservation laws at a junction. For the Cauchy problem, existence, uniqueness, and Lipschitz continuous dependence of the solution from the initial data as well as from the conditions at the junction are proved. The present construction comprehends the case of the p-system used to describe gas flow in networks and hereby unifies different approaches present in the literature. Furthermore, different models for water networks are considered.
TL;DR: In this article, a symmetric Schrodinger/Gross-Pitaeveskii (NLS-GP) with a linear potential was considered and conditions for symmetry-breaking bifurcation were obtained.
Abstract: We consider a class of nonlinear Schrodinger/Gross–Pitaeveskii (NLS-GP) equations, i.e., NLS with a linear potential. NLS-GP plays an important role in the mathematical modeling of nonlinear optical as well as macroscopic quantum phenomena (BEC). We obtain conditions for a symmetry-breaking bifurcation in a symmetric family of states as ${\cal N}$, the squared $L^2$ norm (particle number, optical power), is increased. The bifurcating asymmetric state is a “mixed mode” which, near the bifurcation point, is approximately a superposition of symmetric and antisymmetric modes. In the special case where the linear potential is a double well with well-separation L, we estimate ${\cal N}_{cr}(L)$, the symmetry breaking threshold. Along the “lowest energy” symmetric branch, there is an exchange of stability from the symmetric to the asymmetric branch as ${\cal N}$ is increased beyond ${\cal N}_{cr}$.
TL;DR: If the current is nowhere perpendicular to the boundary or if the minimal current on the boundary, at points where it is perpendicular to it, is greater than the critical current in the one-dimensional case, then the normal state is stable.
Abstract: The stability of the normal state of superconductors in the presence of electric currents is studied in the large domain limit. The model being used is the time-dependent Ginzburg–Landau model, in the absence of an applied magnetic field, and with the effect of the induced magnetic field being neglected. We find that if the current is nowhere perpendicular to the boundary, or if the minimal current on the boundary, at points where it is perpendicular to it, is greater than the critical current in the one-dimensional case, then the normal state is stable. We also prove some short-time instability when the current is both perpendicular to the boundary and smaller than the one-dimensional critical current.
TL;DR: Let $\Omega$ be an open subset of ${\bf R}^n$, where $2\leq n n 7$ and $n\geq 2$ is assumed because the case $n=1$ has been treated elsewhere.
Abstract: Let $\Omega$ be an open subset of ${\bf R}^n$, where $2\leq n\leq 7$; we assume $n\geq 2$ because the case $n=1$ has been treated elsewhere (see [S. S. Alliney, IEEE Trans. Signal Process., 40 (199...
TL;DR: A rigorous foundation for the geometric interpretation of the Hunter–Saxton equation as the equation describing the geodesic flow of the $\dot{H}^1$ right-invariant metric on the quotien...
Abstract: We provide a rigorous foundation for the geometric interpretation of the Hunter–Saxton equation as the equation describing the geodesic flow of the $\dot{H}^1$ right-invariant metric on the quotien...
TL;DR: This work derives incompressible e-MHD equations from compressible Euler–Maxwell equations via the quasi-neutral regime using the curl-div decomposition of the gradient and the wave-type equations of the Maxwell equations.
Abstract: We derive incompressible e-MHD equations from compressible Euler–Maxwell equations via the quasi-neutral regime. Under the assumption that the initial data are well prepared for the electric density, electric velocity, and magnetic field (but not necessarily for the electric field), the convergence of the solutions of the compressible Euler–Maxwell equations in a torus to the solutions of the incompressible e-MHD equations is justified rigorously by studies on a weighted energy. One of the main ingredients for establishing uniform a priori estimates is to use the curl-div decomposition of the gradient and the wave-type equations of the Maxwell equations.
TL;DR: The $\Gamma$-limit of the Ohta–Kawasaki density functional theory of diblock copolymers is a nonlocal free boundary problem and a subrange of the parameters is found where the multiple sphere pattern is stable.
Abstract: The -limit of the Ohta-Kawasaki density functional theory of dibloc k copolymers is a nonlocal free boundary problem. For some values of block composition and the nonlocal interaction, an equilibrium pattern of many spheres exists in a three dimensional domain. A sub-range of the parameters is found where the multiple sphere pattern is stable. This stable pattern models the spherical phase in the diblock copolymer morphology. The spheres are approximately round. They satisfy an equation that involves their mean curvature and a quantity that depends nonlocally on the whole pattern. The locations of the spheres are determined via a Green’s function of the domain.
TL;DR: It is proved that the Degasperis–Procesi equation has global strong solutions and these global solutions decay to zero as time goes to infinity provided the potentials associated with their initial data are of one sign.
Abstract: In this paper, we mainly study several problems on the weakly dissipative Degasperis–Procesi equation. We first establish the local well-posedness of the equation, derive a precise blow-up scenario, and present two blow-up criteria for strong solutions to the equation. We then give the precise blow-up rate of blow-up solutions to the equation. We finally prove that the equation has global strong solutions and these global solutions decay to zero as time goes to infinity provided the potentials associated with their initial data are of one sign.
TL;DR: If the nondecreasing initial datum approaches the constant states $u_\pm$ ($u_-
Abstract: In this paper, the large time behavior of solutions of the Cauchy problem for the one-dimensional fractal Burgers equation $u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0$ with $\alpha\in (1,2)$ is studied. It is shown that if the nondecreasing initial datum approaches the constant states $u_\pm$ ($u_-
TL;DR: Using techniques from maximal regularity and heat-kernel estimates, it is proved the existence of a unique solution to systems of reaction-diffusion equations with mixed Dirichlet–Neumann boundary conditions on nonsmooth domains.
Abstract: In this paper we investigate quasilinear systems of reaction-diffusion equations with mixed Dirichlet–Neumann boundary conditions on nonsmooth domains. Using techniques from maximal regularity and heat-kernel estimates we prove the existence of a unique solution to systems of this type.
TL;DR: This work investigates the asymptotic behavior, as $\varepsilon$ tends to $0^+$, of the transverse displacement of a Kirchhoff–Love plate composed of two domains with respect to each other.
Abstract: We investigate the asymptotic behavior, as $\varepsilon$ tends to $0^+$, of the transverse displacement of a Kirchhoff–Love plate composed of two domains $\Omega_\varepsilon^+\cup\Omega^-_\varepsil...